New SSD with Time Machine

Your Time Machine backup drive should be ~ 4 times the size of the disk you are backing up, or even larger. Your new internal drive will look like it has a lot of "new" files to back up to Time Machine, so you can expect the first backup to take a long time and take up a lot of space. I suggest that you "archive" the existing Time Machine disk and get a new, much larger one and start a new Time Machine backup sequence with the new backup disk. A 4 TB external drive should cost less than $100.

The drives are identified by a hardware address, not just by the name that you give the drive. BdAqua provided some links which were written some years ago for much older operating systems, but they provide ways you can try to force Time Machine to inherit the existing backups. I don't know if those methods will work with High Sierra, but they might. Apple recommends that the backup drive be several times larger than the drive being backed up, it's not just to have backups going back ~ years, it is because the normal backup and versioning process can quickly require much more space than you are using on your drive.

I respectfully urge you to allocate your resources (budget-wise) to put very high priority on having a reliable backup process. For computers and associated equipment, it's not a matter of if they will fail, it's just a matter of when they will fail.

Maybe see these...

https://www.baligu.com/pondini/TM/B5.html

https://www.baligu.com/pondini/TM/B6.html